| Version 3 |
Version 2 |
| If $x$ and $y$ are non-negative real numbers, we can form their arithmetic |
If $x$ and $y$ are non-negative real numbers, we can form their arithmetic |
|
mean $a_0 = (x+y)/2$ as well as their geometric mean $g_0 = \sqrt{xy}$.
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mean $a_0 = (x+y)/2$ as well as their geometic mean $g_0 = \sqrt{xy}$.
|
| This procedure can be repeated to form a sequence of arithmetic and |
This procedure can be repeated to form a sequence of arithmetic and |
| geometic means $a_{n+1} = (a_n+g_n)/2$ and $g_{n+1} = \sqrt{a_n g_n}$. |
geometic means $a_{n+1} = (a_n+g_n)/2$ and $g_{n+1} = \sqrt{a_n g_n}$. |
| By the arithmetic-geometric means inequality we have $a_n \ge a_{n+1} \ge g_{n+1} \ge g_n$ (with equality holding only when $a_n=g_n$), |
By the arithmetic-geometric means inequality we have $a_n \ge a_{n+1} \ge g_{n+1} \ge g_n$ (with equality holding only when $a_n=g_n$), |
| hence these sequences converge to a number between $x$ and $y$, |
hence these sequences converge to a number between $x$ and $y$, |
| with the rate of convergence being \PMlinkname{linear}{LinearConvergence}. |
with the rate of convergence being \PMlinkname{linear}{LinearConvergence}. |
| The \emph{arithmetic-geometric mean} $M(x,y)$ of $x$ and $y$ |
The \emph{arithmetic-geometric mean} $M(x,y)$ of $x$ and $y$ |
| is defined as this limit |
is defined as this limit |
| \begin{equation*} |
\begin{equation*} |
| M(x,y) = \lim_{n\to\oo} a_n, g_n. |
M(x,y) = \lim_{n\to\oo} a_n, g_n. |
| \end{equation*} |
\end{equation*} |
| The origin of the name is obvious from the construction. Alternative notations |
The origin of the name is obvious from the construction. Alternative notations |
| for $M(x,y)$ are $\agm(x,y)$ or $\AGM(x,y)$. |
for $M(x,y)$ are $\agm(x,y)$ or $\AGM(x,y)$. |
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| The AGM lies between the arithmetic and geometric |
The AGM lies between the arithmetic and geometric |
| means of $x$ and $y$, |
means of $x$ and $y$, |
| \begin{equation*} |
\begin{equation*} |
| \frac{x+y}{2} \ge M(x,y) \ge \sqrt{xy}, |
\frac{x+y}{2} \ge M(x,y) \ge \sqrt{xy}, |
| \end{equation*} |
\end{equation*} |
| with equality holding only in case of equality $x=y$. The AGM is also a |
with equality holding only in case of equality $x=y$. The AGM is also a |
| homogeneous function of degree $1$, namely $M(\alpha x, \alpha y) |
homogeneous function of degree $1$, namely $M(\alpha x, \alpha y) |
| = \alpha M(x,y)$ for $\alpha > 0$. It is also symmetric $M(x,y) = M(y,x)$. |
= \alpha M(x,y)$ for $\alpha > 0$. It is also symmetric $M(x,y) = M(y,x)$. |
| These properties are obvious from the construction. |
These properties are obvious from the construction. |
|
|
| The AGM can be used to numerically evaluate elliptic integrals of the |
The AGM can be used to numerically evaluate elliptic integrals of the |
| first and second kinds. For example, |
first and second kinds. For example, |
| \begin{equation} |
\begin{equation} |
| M(x,y) = \frac{\pi}{4} \frac{x+y}{K\left(\frac{|x-y|}{x+y}\right)}, |
M(x,y) = \frac{\pi}{4} \frac{x+y}{K\left(\frac{|x-y|}{x+y}\right)}, |
| \end{equation} |
\end{equation} |
| where $K(k)$ is the elliptic integral of the first kind as function of |
where $K(k)$ is the elliptic integral of the first kind as function of |
| the modulus $k$. |
the modulus $k$. |
